STANDARD SURFACES AND NODAL CURVES IN SYMPLECTIC 4-MANIFOLDS Michael Usher Abstract

77(2007) 237-290

STANDARD SURFACES AND NODAL CURVES IN SYMPLECTIC 4-MANIFOLDS

Michael Usher

Abstract

Continuing the program of [DS] and [U1], we introduce re- finements of the Donaldson-Smith standard surface count which are designed to count nodal pseudoholomorphic curves and curves with a prescribed decomposition into reducible components. In cases where a corresponding analogue of the Gromov-Taubes in- variant is easy to define, our invariants agree with those analogues.

We also prove a vanishing result for some of the invariants that count nodal curves.

1. Introduction

Let (X, ω) be a closed symplectic 4-manifold. We assume that [ω]∈ H²(X,Z); however, the main theorems in this paper concern Gromov invariants, which are unchanged under deformations of the symplectic form, so since any symplectic form is deformation equivalent to an in- tegral form there is no real loss of generality here. According to [Do], if k is large enough, taking a suitable pair of sections of a line bundle L^⊗k where L has Chern class [ω] and blowing X up at the common vanishing locus of these sections to obtain the new manifold X^′ gives rise to a symplectic Lefschetz fibration f: X^′ → CP¹ (the exceptional curves of the blowup π: X^′ → X appear as sections of f, while at other points x^′ ∈ X^′, f(x^′) ∈ C∪ {∞} is the ratio of the two chosen sections of L^⊗k at π(x^′) ∈ X). In other words, f is a fibration by Riemann surfaces over the complement of a finite set of critical values in S², while near its critical pointsf is given in smooth local complex coordinates by f(z, w) = zw. Results of [Sm1] show that the critical points of f may be assumed to lie in separate fibers, and all fibers of f may be assumed irreducible. Once we choose a metric on X^′, Donald- son’s construction thus presents a suitable blowup of X as a smoothly CP¹-parametrized family of Riemann surfaces, all but finitely of which are smooth and all of which are irreducible with at worst one ordinary double point. Where κ_X = c₁(T^∗X) is the canonical class of X, note

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that the adjunction formula gives the arithmetic genus of the fibers as g= 1 + (k²[ω]²+kκ_X ·ω)/2.

Beginning with the work of S. Donaldson and I. Smith in [DS], some efforts have recently been made toward determining whether such a Lefschetz fibration can shed light on any questions concerning pseudo- holomorphic curves in X. More specifically, for any natural number r Donaldson and Smith construct the relative Hilbert scheme, which is a smooth symplectic manifoldXr(f) with a mapF: Xr(f)→CP¹whose fiber over a regular value tof f is the symmetric product S^rf⁻¹(t). If we choose an almost complex structure j on X^′ with respect to which f is a pseudoholomorphic map, a j-holomorphic curve C in X^′ which contains no fiber components will, by the positivity of intersections be- tweenj-holomorphic curves, meet each fiber inr := [C]·[f iber] points, counted with multiplicities. In other words, C∩f⁻¹(t)∈S^rf⁻¹(t), so that, letting tvary,C gives rise to a sectionsC ofXr(f). Conversely, a section sof X_r(f) gives rise to a subset C_s of X^′ (namely the union of all the points appearing in the divisorss(t) as tvaries), and fromj one may construct a (nongeneric and generally not even C¹) almost com- plex structureJ_j with the property that C is a (possibly disconnected) j-holomorphic curve in X^′ if and only ifs_C is aJ_j-holomorphic section of X_r(f).

Accordingly, it seems reasonable to study pseudoholomorphic curves in X^′ by studying pseudoholomorphic sections of Xr(f). If α ∈ H²(X^′;Z), thestandard surface count DS_f(α) is defined in [Sm2] (and earlier in [DS] for α = κ_X′) as the Gromov-Witten invariant which countsJ-holomorphic sectionsswhose corresponding setsC_s are Poin- car´e dual to the classαand pass through a generic set ofd(α) = ¹₂(α²− κ_X′ ·α) points of X^′, where J is a generic almost complex structure on X_r(f). Smith shows in [Sm2] that there is at most one homotopy class c_α of sections ssuch that C_s is Poincar´e dual toα, and moreover that the complex dimension of the space of J-holomorphic sections in this homotopy class is, for generic J, the aforementioned d(α), which the reader may recognize as the same as the expected dimension of j- holomorphic submanifolds ofXPoincar´e dual toα. Further, the moduli space ofJ-holomorphic sections in the homotopy classc_αis compact for genericJifkis taken large enough. The moduli space in the definition of DS_f is therefore a finite set, andDS_f simply counts the members of this set with sign according to the usual (spectral-flow-based) prescription.

Donaldson and Smith have proven various results aboutDS, perhaps the most notable of which is the main theorem of [Sm2], which asserts that if α ∈H²(X;Z), if b⁺(X)> b₁(X) + 1, and if the degreek of the Lefschetz fibration is high enough, then

(1.1) DS_f(π^∗α) =±DS_f(π^∗(κ_X −α)).

Their work has led to new, more symplectic proofs of various results in 4-dimensional symplectic topology which had previously been accessi- ble only by Seiberg-Witten theory (as an example we mention the main theorem of [DS], according to which X admits a symplectic surface Poincar´e dual to κ_X, again assuming b⁺(X) > b1(X) + 1). In [U1] it was shown that the invariantDS_f agrees with the Gromov invariantGr which was introduced by C. Taubes in [T2] and which counts possibly- disconnected pseudoholomorphic submanifolds ofX^′ Poincar´e dual to a given cohomology class. This in particular shows that DS_f is indepen- dent of the choice of Lefschetz fibration structure, and, in combination with Smith’s duality theorem (1.1) and the fact that under a blowup π one hasGr(π^∗α) =Gr(α), yields a new proof of the relation

Gr(α) =±Gr(κ_X −α)

ifb⁺(X)> b1(X) + 1, a result which had previously only been known as a shadow of the charge conjugation symmetry in Seiberg-Witten theory.

The information contained in the Gromov invariants comprises only a part of the data that might be extracted from pseudoholomorphic curves inX. The present paper aims to show that many of these addi- tional data can also be captured by Donaldson-Smith-type invariants.

For instance, Gr(α) counts all of the curves with any decomposition into connected components whose homology classes add up (counted with multiplicities) to α. It is natural to wish to keep track of the de- compositions of our curves into reducible components; accordingly we make the following:

Definition 1.1. Letα∈H²(X;Z). Let

α=β1+· · ·+βm+c1τ1+· · ·+cnτn

be a decomposition of α into distinct summands, where none of theβ_i satisfies β_i² = κX ·βi = 0, while the τi are distinct classes which are primitive in the latticeH²(X;Z) and all satisfyτ_i²=κ_X ·τ_i= 0. Then

Gr(α;β1, . . . , βm, c1τ1,· · · , cnτn)

is the invariant counting ordered (m+n)-tuples (C₁, . . . , C_m+n) of trans- versely intersecting smooth pseudoholomorphic curves inX, where

(i) for 1≤i≤m,C_i is a connected curve Poincar´e dual to β_i which passes through some prescribed generic set of d(β_i) points;

(ii) form+ 1≤k≤m+n,C_kis a union of connected curves Poincar´e dual to classes l_k,1τ_k,· · · , l_k,pτ_k decorated with positive integer multiplicities m_k,q with the property that

X

q

mk,qlk,q =ck.

The weight of each component of each such curve is to be determined according to the prescription given in the definition of the Gromov in- variant in [T2] (in particular, the components C_k,q in class l_k,qτ_k are given the weight r(C_k,q, m_k,q) specified in Section 3 of [T2]), and the contribution of the entire curve is the product of the weights of its com- ponents.

The objects counted byGr(α;α₁, ..., α_n) will then be reducible curves with smooth irreducible components and a total ofP

α_i·α_jnodes arising from intersections between these components. Gr(α) is the sum over all decompositions of α into classes which are pairwise orthogonal under the cup product of the

d(α)!

Q(d(α_i)!)Gr(α;α₁, . . . , α_n);

in turn, one has

Gr(α;α₁, . . . , α_n) = Yn i=1

Gr(α_i;α_i).

In Section 2, given a symplectic Lefschetz fibration f: X → S² with sufficiently large fibers, by counting sections of a relative Hilbert scheme we construct a corresponding invariant gDSf(α;α1, . . . , αn) pro- vided that none of the α_i can be written as mβ where m >1 and β is Poincar´e dual to either a symplectic square-zero torus or a symplectic (−1)-sphere. Further:

Theorem 1.2. ^(QP_(d(α^d(αi))!

i)!)Gr(α;α₁, . . . , α_n) =DSg_f(α;α₁, . . . , α_n)pro- vided that the degree of the fibration is large enough that h[ω_X],[Φ]i >

[ω_X]·α.

The sections s counted by gDS_f(α;α₁, . . . , α_n) correspond tautologi- cally to curves Cs =∪C_sⁱ inX with each C_sⁱ Poincar´e dual to αi. The C_sⁱ will be symplectic, and Proposition 2.5 guarantees that they will intersect each other positively, so there will exist an almost complex structure making C_s holomorphic. However, ifs₁ and s₂ are two differ- ent sections in the moduli space enumerated by DSg_f(α;α₁, . . . , α_n), it is unclear whether there will exist a single almost complex structure on X making both C_s1 and C_s2 holomorphic.

The almost complex structures on X_r(f) used in the definition of gDS are, quite crucially, required to preserve the tangent space to the diagonal stratum consisting of divisors with one or more points re- peated. One might hope to define analogous invariants which agree with Gr(α;α₁, . . . , α_n) using arbitrary almost complex structures on X_r(f). If one could do this, though, the arguments reviewed in Section 4 would rather quickly enable one to conclude thatGr(α;α₁, . . . , α_n) = 0

whenever α has larger pairing with the symplectic form than does the canonical class and α_i·α_j = 0 for i6=j. However, this is not the case:

the manifold considered in [MT] admits a symplectic form such that, for certain primitive, orthogonal, square-zero classesα,β,γ, andδeach with positive symplectic area, the canonical class is 2(α+β +γ) but the invariant Gr(2(α+β+γ) +δ;α, β, γ, α+β+γ+δ) is nonzero.

While the Gromov–Taubes invariant restricts attention to curves whose components are all covers of embedded curves which do not in- tersect each other, it is natural to hope for information about curves Poincar´e dual toαhaving some numbernof transverse self-intersections.

One might like to define an analogueGrn(α) of the Gromov–Taubes in- variant counting such curves, but as we review in Section 3, owing to issues relating to multiple covers it is somewhat unclear what the def- inition of such an invariant should be in general. If one imposes some rather stringent conditions on α (α should be “n-semisimple” in the sense of Definition 3.1), there is however a natural such choice.

Note that for arbitrary α and n, following [RT] one may define an invariantRT_n(α) which might naively be viewed as a count ofconnected pseudoholomorphic curves Poincar´e dual to α with n self-intersections by enumerating solutionsu: Σg→Xof the equation ( ¯∂ju) =ν(x, u(x)) for generic j and “inhomogeneous term” ν, where the genus g of the source curve is given in accordance with the adjunction formula by 2g− 2 = α² +κ_X ·α −2n. (Note that the nontrivial dependence of ν on x prevents multiple cover problems from arising.) In the case n = 0, the main theorem of [IP1] provides a universal formula equatingGr(α) with a certain combination of the Ruan–Tian invariantsRT₀. The proof of that theorem goes through easily to show that in the case when α is n-semisimple, there exists a similar formula equating Gr_n(α) with a combination of Ruan–Tian invariants. We mention also that, again as an artifact of the multiple cover problem, the Ruan–Tian invariants are obliged to take values inQrather thanZ. Gr(α), on the other hand, is an integer-valued invariant.

By combining the approaches of [DS] and [L1], in the presence of a Lefschetz fibration f: X → S² we construct in Section 3 an integer- valued invariantFDSⁿ_f(α−2P

e_i) which we conjecture to be an appro- priate candidate for a “nodal version” Gr_n(α) of the Gromov invariant for general classesα(after perhaps dividing by n! to account for a sym- metry in the construction). Pleasingly, the technical difficulties that often arise in defining invariants like Gr_n(α) do not affect FDS: since FDS counts sections of a (singular) fibration, which of course neces- sarily represent a primitive homology class in the total space, we need not worry about multiple covers; further, the fact that any bubbles that form in the limit of a sequence of holomorphic sections must be con- tained in the fibers of the fibration turns out (via an easy elaboration

of a dimension computation from [DS]) to generically rule out bubbling as well. In principle, though, FDSⁿ_f might depend on the choice of Lefschetz fibration f.

Note that ifπ: X^′ →Xis a blowup with exceptional divisor Poincar´e dual to ǫ, whenever Gr_n(β) is defined we will have (Gr_n)_X′(β+ǫ) = (Gr_n)_X(β) (here and elsewhere we use the same notation for β ∈ H²(X;Z) and π^∗β ∈ H²(X^′;Z)), as the curves contributing to (Gr_n)_X(β) generically miss the point being blown up, and so the unions of their proper transforms with the exceptional divisor will be precisely the curves contributing to (Gr_n)_X′(β+ǫ). With this said, we formulate:

Conjecture 1.3. Let (X, ω) be a symplectic 4-manifold and α ∈ H²(X;Z), andf: X^′ →S² a Lefschetz fibration obtained from a suffi- ciently high-degree Lefschetz pencil onX, with the exceptional divisors of the blowup X^′ → X Poincar´e dual to the classes ǫ₁, . . . , ǫ_N. Then the family Donaldson–Smith invariants

FDSⁿ_f Ã

α+ XN

i=1

ǫ_i−2 Xn k=1

e_k

!

are independent of the choice of f, and have a general expression in terms of the Ruan–Tian invariants of X.

Note that this conjectural general expression would then produce an integer by taking appropriate combinations of the (a priori only rational) Ruan–Tian invariants, similarly to the formula of [IP1].

In light of the behavior ofGr_nunder blowups, Theorem 3.8 amounts to the statement that:

Theorem 1.4. If α is strongly n-semisimple, then Conjecture 1.3 holds for α; more specifically, we have

FDSⁿ_f Ã

α+ XN

i=1

ǫ_i−2 Xn k=1

e_k

!

=n!Gr_n(α).

We also prove that FDS vanishes under certain circumstances. This result depends heavily on the constructions used by Smith in [Sm2]

to prove his duality theorem, and so we review these constructions in Section 4. Section 5 is then devoted to a proof of the following theorem.

Theorem 1.5. Ifb⁺(X)> b₁(X) + 4n+ 1, then for allα ∈H²(X;Z) such that r = hα,[Φ]i satisfies r > max{g(Φ) + 3n+d(α),(4g(Φ)− 11)/3}, either FDSⁿ_f(α−2P

e_i) = 0 or there exists an almost complex structure j on X compatible with the fibration f: X → S² which si- multaneously admits holomorphic curves C and DPoincar´e dual to the classes α and κX −α. In particular, FDSⁿ_f(α−2P

ei) = 0 if α has larger pairing with the symplectic form than does κ_X.

Note that in the Lefschetz fibrations obtained from degree-kLefschetz pencils on some fixed symplectic manifold (X, ω), the number N of exceptional sections is k²[ω]² while the number 2g(Φ)−2 is asymptotic tok²[ω]², so the invariants

FDSⁿ_f Ã

α+ XN

i=1

ǫi−2X

k

e_k

!

considered in Conjecture 1.3 all eventually satisfy the restriction on r in Theorem 1.5.

The almost complex structure in the second alternative in Theorem 1.5 cannot be taken to be regular (in the sense that the moduli spaces M^j_X(β) ofj-holomorphic curves Poincar´e dual toβ are of the expected dimension); the most we can say appears to be that it can be taken to be a member of a regular 4n-real-dimensional family of almost complex structures,i.e., a family of almost complex structures{j_b}parametrized by elements b of an open set in R⁴ⁿ such that the spaces {(b, C)|C ∈ M^j_X^b(β)}are of the expected real dimension 2d(β) + 4nnear each (b, C) such that C has no multiply-covered components. Also, if X is in fact K¨ahler and admits a compatible integrable complex structure j₀ with respect to which the fibration f is holomorphic, then we can take thej in Theorem 1.5 equal to j₀.

In fact, if we could take j to be regular, then we could rule out the second alternative in Theorem 1.5 entirely (when n >0) using the following argument: the invariant vanishes trivially when d(α) < n, so we can assume d(α) = −¹₂α ·(κ−α) > 0. But then our curves Poincar´e dual to αand κ−αhave negative intersection number, which is only possible if they share one or more components of negative square.

For generic j, a virtual dimension computation shows that the only irreducible j-holomorphic curves of negative square are (−1)-spheres.

Moreover whatever (−1)-spheres appear in X must be disjoint, since if they were not, blowing one of two intersecting (−1)-spheres down would cause the image of the other to be a symplectic sphere of nonnegative self-intersection, which (by a result of [McD]) would force X to have b⁺ = 1, which we assumed it did not. Ignoring all the (−1)-spheres in C and D and taking the union of what is left over would then give a j-holomorphic curve Poincar´e dual to a class κ_X −P

a_ie_i where the e_i are classes of (−1)-spheres with e_i·e_k = 0 for i 6= k and where at least oneai≥2. But one easily finds d(κ_X −P

aiei)<0, so this too is impossible for generic j. For nongeneric j, this argument breaks down because of the possibility that C and D might share components of negative square and negative expected dimension, and there is a wider diversity of possible homology classes of such curves.

The final section of the paper contains proofs of two technical results that are used in the proofs of the main theorems. First, we show that

the operation of blowing up a point can be performed in the almost complex category, a fact which does not seem to appear in the literature and whose proof is perhaps more subtle than one might anticipate. The paper then closes with a proof of the following result, which is necessary for the compactness argument that we use to justify the definition of our invariant FDS:

Theorem 1.6. Let F: Hr → D² denote the r-fold relative Hilbert scheme of the map(z, w)7→zw,φ₀the partial resolution mapF⁻¹(0)→ Sym^r{zw = 0}, and ∆ ⊂ H_r the diagonal stratum. At any point p ∈

∆∩F⁻¹(0) with φ₀(p) ={(0,0), . . . ,(0,0)}, where T_p∆ is the tangent cone to ∆ at p, we have Tp∆⊂TpF⁻¹(0).

We end the introduction with some remarks on the possible relation of FDS to (family) Seiberg–Witten theory. In [Sa] it was shown that where X is the product of R and a fibered three-manifold, so that X fibers over a cylinder, if one examines the Seiberg–Witten equations on Xusing a family of metrics for which the size of the fibers shrinks to zero, then one obtains in the adiabatic limit the equations for a holomorphic family of solutions to the symplectic vortex equations on the fibers. In turn, there is a natural isomorphism between the space of solutions to the vortex equations on a Riemann surface and the symmetric product of the surface. In other words, in this simple context the adiabatic limit of the Seiberg–Witten equations is the equation for a holomorphic family of elements of the symmetric products of the fibers of the fibrationX→ R×S¹. As was noted in [DS], since for a Lefschetz fibration f: X→ S² DS_f precisely counts pseudoholomorphic families of elements of the symmetric products of the fibers of f, one might take inspiration from Salamon’s example and hope to obtain the equivalence between DS_f and the Seiberg–Witten invariant by considering the Seiberg–Witten equations onX for a family of metrics with respect to which the size of the fibers shrinks to zero.

Now our invariant FDSⁿ_f is constructed by counting pseudoholo- morphic families of elements of the symmetric products of the fibers of a family of Lefschetz fibrations f^b obtained by restricting a map fn: Xn+1→ S²×Xn to the preimage X^b ofS²× {b} asb ranges over the complementX_n^′ of a set of codimension 4 inX_n. In the above vein, one might hope to relate the family Seiberg–Witten invariants F SW for the family of 4-manifolds X_n+1 → X_n (which enumerate Seiberg–

Witten monopoles in the various X^b as b ranges over X_n; see, e.g., [LL]) to FDSⁿ_f via an adiabatic limit argument. This would in partic- ular yield a proof of the independence of FDSⁿ_f from f in Conjecture 1.3, and indeed may well be the most promising way to establish this independence in the absence of a suitable invariantGr_n (or of a “family Gromov–Taubes invariant” F Gr) with whichFDSⁿ_f might be directly equated.

As was shown in [L2], whenX is an algebraic surface andb⁺(X) = 1 the family Seiberg–Witten invariants agree with certain curve counts in algebraic geometry. For larger values ofb⁺, though, the family Seiberg–

Witten invariants that are hoped to correspond to nodal curve counts are expected to vanish due to the fact that symplectic manifolds have Seiberg–Witten simple type; note that Theorem 1.5 suggests thatFDSⁿ_f also tends to vanish for large b⁺. By contrast, there are plenty of non- trivial nodal curve counts in algebraic surfaces withb⁺>1 (see [L1] for a review of some of these); these counts correspond to Liu’s “algebraic Seiberg–Witten invariants” ASW and differ from F SW when b⁺>1.

Acknowledgements.Section 2 of this paper appeared in my thesis [U2]; I would like to thank my advisor Gang Tian for suggesting that I attempt to study nodal curves using the Donaldson–Smith approach.

Thanks also to Cliff Taubes for helping me identify an error an an earlier version of this paper, to Dusa McDuff for making me aware of the need for Section 6.1, and to Ivan Smith for helpful remarks. This work was partially supported by the National Science Foundation.

2. Refining the standard surface count

Throughout this section,Xr(f) will denote the relative Hilbert scheme constructed from some high-degree but fixed Lefschetz fibrationf: X → S² obtained by Donaldson’s construction applied to the fixed symplec- tic 4-manifold (X, ω). The fiber of f over t ∈ S² will occasionally be denoted by Σ_t, and the homology class of the fiber by [Φ].

As has been mentioned earlier, DS_f(α) is a count of holomorphic sections of the relative Hilbert scheme X_r(f) in a certain homotopy class cα characterized by the property that if sis a section in the class c_α then the closed setC_s⊂X“swept out” bys(that is, the union over all tof the divisorss(t)∈Σ_t) is Poincar´e dual toα(note that points of C_sin this interpretation may have multiplicity greater than 1). Thatc_α is the unique homotopy class with this property is seen in Lemma 4.1 of [Sm2]; in particular, for instance, we note that sections which descend toconnected standard surfaces Poincar´e dual toαare not distinguished at the level of homotopy from those which descend to disjoint unions of several standard surfaces which combine to representP D(α).

Of course, in studying standard surfaces it is natural to wish to know their connected component decompositions, so we will presently attempt to shed light on this. Suppose that we have a decomposition

α=α1+· · ·+αn

with

hα,[Φ]i=r, hα_i,[Φ]i=r_i.

Over each t∈S² we have an obvious “divisor addition map”

+ : Yn i=1

S^riΣ_t→S^rΣ_t

(D₁, . . . , D_n)7→D₁+· · ·+D_n; allowingt to vary we obtain from this a map on sections:

+ : Yn i=1

Γ(Xri(f))→Γ(Xr(f)) (s₁, . . . , s_n)7→

Xn i=1

s_i. As should be clear, one has

+(c_α1 × · · · ×c_αn)⊂c_α if α=P

αi, since C^P_αi is the union of the standard surfaces Csi and hence is Poincar´e dual to α if eachC_si is Poincar´e dual to α_i. Further, we readily observe:

Lemma 2.1. The image+(c_α1×· · ·×c_αn)⊂c_α is closed with respect to the C⁰ norm.

Proof. Suppose we have a sequence (s^m₁ , . . . , s^m_n)^∞_m=1inc_α1× · · · ×c_αn such that P

s^m_i → s ∈ c_α. Now each S^riΣ_t is compact, so at each t, each of the sequencess^m_i (t) must have subsequences converging to some s⁰_i(t). But then necessarily each P

s⁰_i(t) = s(t), and then we can see by, for anyl, fixing the subsequence used for alli6=l and varying that used for i=l that in fact every subsequence ofs^m_l (t) must converge to s⁰_l(t). Lettingtvary then gives sectionss⁰_i such that everys^m_i →s⁰_i and Ps⁰_i =s; the continuity of sis readily seen to imply that of the s⁰_i. q.e.d.

At this point it is useful to record an elementary fact about the lin- earization of the divisor addition map.

Proposition 2.2. Let Σ be a Riemann surface and r = P

ri. The linearization +_∗ of the addition map

+ : Yn i=1

S^riΣ→S^rΣ

at(D₁, . . . , D_n)is an isomorphism if and only ifD_i∩D_j =∅for i6=j.

If two or more of the D_i have a point in common, then the image of+_∗ at(D1, . . . , Dn) is contained in T^P_Di∆, where∆⊂S^rΣis the diagonal stratum consisting of divisors with a repeated point.

Proof. By factoring + as a composition

S^r1Σ×S^r2Σ× · · · ×S^rnΣ→S^r1+r2Σ× · · · ×S^rnΣ→ · · · →S^rΣ in the obvious way we reduce to the case n= 2. Now in general for a divisor D = P

a_ip_i ∈ S^dΣ where the p_i are distinct, a chart for S^dΣ is given by Q

S^aiU_i, where the U_i are holomorphic coordinate charts around p_i and the S^aiU_i use as coordinates the elementary symmetric polynomialsσ₁, . . . , σ_ai in the coordinates ofU_i^ai. As such, ifD₁andD₂ are disjoint, a chart around D₁+D₂∈S^r1+r2Σ is simply the Cartesian product of charts around D₁ ∈ S^r1Σ and D₂ ∈ S^r2Σ, and the map + takes the latter diffeomorphically (indeed, biholomorphically) onto the former, so that (+_∗)_(D1,D2) is an isomorphism.

On the other hand, note that

+ : S^aC×S^bC→S^a+bC

is given in terms of the local elementary symmetric polynomial coordi- nates around the origin by

(σ₁, . . . , σ_a, τ₁, . . . , τ_b)7→(σ₁+τ₁, σ₂+σ₁τ₁+τ₂, . . . , σ_aτ_b), and so has linearization

(+∗)_{(σ1,...,τb)}(η1, . . . , ηa, ζ1, . . . , ζ_b)

= (η₁+ζ₁, η₂+σ₁ζ₁+τ₁η₁+ζ₂, . . . , σ_aζ_b+τ_bη_a).

We thus see that Im(+_∗)_(0,...,0) only has dimension max{a, b} and is contained in the image of the linearization of the smooth model

C×S^a+b−2C→S^a+bC (z, D)7→2z+D

for the diagonal stratum at (0,0+· · ·+0). Suppose now thatD1andD2

contain a common pointp; writeD_i =a_ip+D_i^′ whereD_i∈S^ri−aiΣ are divisors which do not containp. Then from the commutative diagram

S^a1Σ×S^r1−a1Σ×S^a2Σ×S^r2−a2Σ −−−−→ S^r1Σ×S^r2Σ

 y

 y⁺ S^a1+a2Σ×S^{r1+r2−a1−a2}Σ −−−−→ S^r1+r2Σ

and the fact that the linearization of the top arrow at (a₁p, D₁^′, a₂p, D₂^′) is an isomorphism (by what we showed earlier, since the D_i^′ do not contain p), while the linearization of the composition of the left and bottom arrows at (a₁p, D₁^′, a₂p, D₂^′) has image contained in T_D1+D2∆, it follows that (+_∗)_(D1,D2) has image contained in T_D1+D2∆ as well, which suffices to prove the proposition. q.e.d.

Corollary 2.3. Ifs_i ∈Γ(X_ri(f))are differentiable sections such that C_si∩C_sj 6=∅ for some i6=j, then s=P

s_i ∈Γ(X_r(f)) is tangent to the diagonal stratum of X_r(f).

Proof. Indeed, if C_si∩C_sj 6= ∅, then there is x ∈ S² such that the divisorss_i(x) ands_j(x) contain a point in common, and so forv∈T_xS² we have

s_∗v= (+◦(s_i, s_j))_∗v= +_∗(s_1∗v, s_2∗v)∈T_s(t)∆

by Proposition 2.2. q.e.d.

Note that it is straightforward to find cases in which thesi are only continuous with some C_si ∩C_sj nonempty and the sum s = P

s_i is smooth but not tangent to the diagonal. For example, letr = 2, and in local coordinates let s₁ be a square root of the functionz7→Re(z) and s₂ =−s₁. Then in the standard coordinates on the symmetric product we have s(z) = (0,−Re(z)), so thatT(Ims) shares only one dimension with T∆ at z= 0. If s istransverse to ∆, one can easily check that a similar situation cannot arise.

We now bring pseudoholomorphicity in the picture. Throughout this treatment, all almost complex structures on X_r(f) will be assumed to agree with the standard structures on the symmetric product fibers, to make the map F: X_r(f) → S² pseudoholomorphic, and, on some (not fixed) neighborhood of the critical fibers of F, to agree with the holomorphic model for the relative Hilbert scheme over a disc around a critical value for f provided in Section 3 of [Sm2]. Let J denote the space of these almost complex structures. It follows by standard argu- ments (see Proposition 3.4.1 of [MS1] for the general scheme of these arguments and Section 4 of [DS] for their application in the present context) that for genericJ ∈ J the spaceM^J(c_α) is a smooth manifold of (real) dimension 2d(α) = α²−κX ·α (the dimension computation comprises Lemma 4.3 of [Sm2]); this manifold is compact, for bubbling is precluded by the arguments of Section 4 of [Sm2] assuming we have taken a sufficiently high-degree Lefschetz fibration.

InsideM^J(c_α) we have the setM^J(c_α1×· · ·×c_αn) consisting of holo- morphic sections which lie in the image +(c_α1× · · · ×c_αn). By Lemma 2.1 and the compactness of M^J(c_α), M^J(c_α1 × · · · ×c_αn) is evidently compact; however, the question of its dimension or even whether it is a manifold appears to be a more subtle issue in general.

Let us pause to consider what we would like the dimension ofM^J(cα1×

· · · × c_αn) to be. The objects in M^J(c_α1 × · · · ×c_αn) are expected to correspond in some way to unions of holomorphic curves Poincar´e dual to α_i. Accordingly, assume we have chosen theα_i so that d(α_i) =

1

2(α²_i−κ_X·α_i)≥0 (for otherwise we would expectM^J(c_α1×· · ·×c_αn) to be empty). Holomorphic curves in these classes will intersect positively as long as they do not share any components of negative square; for a generic almost complex structure the only such components that can

arise are (−1)-spheres, so if we choose the α_i to not share any (−1)- sphere components (i.e., if the α_i are chosen so that there is no classE represented by a symplectic (−1)-sphere such thathα_i, Ei<0 for more than one α_i), then it would also be sensible to assume that α_i·α_j ≥0 fori6=j.

The above naive interpretation ofM^J(c_α1× · · · ×c_αn) would suggest that its dimension ought to be P

d(αi). Note that d(α) =d(X

α_i) =X

d(α_i) +X

i>j

α_i·α_j,

so under the assumptions on theα_ifrom the last paragraph we have that the expected dimension of M^J(c_α1 × · · · ×c_αn) is at most the actual dimension of M^J(c_α) (as we would hope, given that the former is a subset of the latter), with equality if and only if α_i·α_j = 0 whenever i6=j.

As usual, we will find it convenient to cut down the dimensions of our moduli spaces by imposing incidence conditions, so we shall fix a set Ω of pointsz∈Xand consider the spaceM^J,Ω(c_α1×· · ·×c_αn) of elements s∈ M^J(c_α1× · · · ×c_αn) such thatC_s passes through each of the points z (or, working more explicitly inX_r(f), such thats meets each divisor z+S^r−1Σ_t, Σ_t being the fiber which contains z). M^J,Ω(c_α) is defined similarly, and standard arguments show that for generic choices of Ω M^J,Ω(c_α) will be a compact manifold of dimension

2(d(α)−#Ω).

We wish to count J-holomorphic sections s of X_r(f) such that the reducible components of C_s are Poincar´e dual to the α_i. If we impose Pd(αi) incidence conditions, then according to the above discussion M^J,Ω(c_α) will be a smooth manifold of dimension 2P

i>jα_i ·α_j. A section P

s_i ∈ +(c_α1 × · · ·c_αn) whose summands are all differentiable would then, by Corollary 2.3, have one tangency to the diagonal ∆ for each of the intersections between the C_si, of which the total expected number is P

i>jαi ·αj. This suggests that the sections we wish to count should be found among those elements of M^J,Ω(c_α) which have P

i>jαi·αj tangencies to ∆, where Ω is a set of P

d(αi) points.

To count pseudoholomorphic curves tangent to a symplectic subvari- ety it is necessary to restrict to almost complex structures which pre- serve the tangent space to the subvariety (see [IP2] for the general the- ory when the subvariety is a submanifold). Accordingly, we shall restrict attention to the class of almost complex structures J on X_r(f) which arecompatible with the stratain the sense to be explained presently (for more details, see Section 6 of [DS], in which the notion was introduced).

Within ∆, there are various strata χ_π indexed by partitions π :r = Pa_in_i with at least onea_i >1; these strata are the images of the maps

p_χ: X_n1(f)×_S2· · · ×_S2 X_nk(f)→X_r(f) (D₁, . . . , D_k)7→X

a_iD_i;

in particular, ∆ = χr=2·1+1·(r−2). An almost complex structure J on X_r(f) is said to be compatible with the strata if the mapsp_χare (J^′, J)- holomorphic for suitable almost complex structuresJ^′ on their domains.

Denoting by Yχ the domain of pχ, Lemma 7.4 of [DS] and the dis- cussion preceding it show:

Lemma 2.4([DS]). For almost complex structuresJ onX_r(f)which are compatible with the strata, each J-holomorphic section s of X_r(f) lies in some unique minimal stratum χ and meets all strata contained in χ in isolated points. In this case, there is a J^′-holomorphic sec- tion s^′ of Y_χ such that s=p_χ◦s^′. Furthermore, for generic J among those compatible with the strata, the actual dimension of the space of all such sections s is equal to the expected dimension of the space of J^′-holomorphic sectionss^′ lying over s.

We note the following analogue for standard surfaces of the positivity of intersections of pseudoholomorphic curves.

Proposition 2.5. Let s = m1s1 +· · ·+m_ks_k be a J-holomorphic section ofX_r(f), where thes_i ∈c_αi ⊂Γ(X_ri(f))are each not contained in the diagonal stratum of Xri(f), and where the almost complex struc- ture J on X_r(f) is compatible with the strata. Assume that the s_i are all differentiable. Then all isolated intersection points of C_si and C_sj contribute positively to the intersection number α_i·α_j.

Proof. We shall prove the lemma for the casek= 2, the general case being only notationally more complicated. The analysis is somewhat easier if the points of C_s1 ∩C_s2 ⊂ X at issue only lie over t ∈ S² for which s1(t) and s2(t) both miss the diagonal of Xr1(f) and Xr2(f), respectively, so we first argue that we can reduce to this case. Let χ be the minimal stratum (possibly all of X_r(f)) in which s = m₁s₁ + m₂s₂ is contained, so that all intersections of s with lower strata are isolated. Let p ∈ X be an isolated intersection point of C_s1 and C_s2 lying over 0 ∈ S², and let δ > 0 be small enough that there are no other intersections of swith any substrata ofχ(and so in particular no other points ofCs1∩Cs2) lying overD2δ(0)⊂S². We may then perturb s=m₁s₁+m₂s₂ to ˜s=m₁s˜₁+m₂s˜₂, still lying inχ, such that

(i) Over D_δ(0), ˜s is J-holomorphic and disjoint from all substrata having real codimension larger than 2 inχ, and the divisors ˜s₁(0) and ˜s₂(0) both still containp;

(ii) Over the complement ofD_2δ(0), ˜sagrees with s; and

(iii) Over D_2δ(0)\ D_δ(0), ˜s need not be J-holomorphic but is con- nected to s by a family of sections s_t contained inχ which miss all substrata of χ

(it may be necessary to decrease δ to find such ˜s, but after doing so such ˜swill exist by virtue of the abundance of J-holomorphic sections over the small disc D_δ(0) which are close to s|_Dδ(0)). The contribution of p to the intersection number α₁·α₂ will then be equal to the total contribution of all the intersections of C_s˜1 and C_s˜2 lying over D_δ(0), and the fact that ˜smisses all substrata with codimension larger than 2 inχis easily seen to imply that these intersections (of which there is at least one, at p) are all at points where ˜s₁ and ˜s₂ miss the diagonals in X_r1(f) andX_r2(f).

As such, it suffices to prove the lemma for intersection points at which s₁ and s₂ both miss the diagonal. In this case, in a coordinate neighborhood U aroundp, theC_si can be written as graphsC_si ∩U = {w = g_i(z)}, where w is the holomorphic coordinate on the fibers of X, z is the pullback of the holomorphic coordinate on S², and g_i is a differentiable complex-valued function which vanishes at z = 0.

Suppose first that m₁ = m₂ = 1. Then near s(0), we may use co- ordinates (z, σ1, σ2, y3, . . . , yr) for Xr(f) obtained from the splitting T₀S² ⊕ T_2pS²Σ₀ ⊕T_s(t)−2pS^r−2Σ₀, and the first two vertical coordi- nates of s(z) = (s₁ +s₂)(z) with respect to this splitting are (g₁(z) + g2(z), g1(z)g2(z)). NowsisJ-holomorphic and meets theJ-holomorphic diagonal stratum ∆ at (0, s(0)), and at this point ∆ is tangent to the hy- perplane σ₂ = 0, so it follows from Lemma 3.4 of [IP2] that the Taylor expansion of g₁(z)g₂(z) has form a₀z^d+O(d+ 1). But then the Tay- lor expansions of g₁(z) and g₂(z) begin, respectively, a₁z^d1+O(d₁+ 1) and a₂z^d2 +O(d₂+ 1), withd₁+d₂ =d. Then since C_si ∩U ={w = g_i(z)}, it follows immediately that theC_si have intersection multiplicity max{d1, d2}>0 at p.

There remains the case where one or both of them_i is larger than 1.

In this case, where Yχ =Xr1(f)×_S² Xr2(f) is the smooth model for χ, becauseJ is compatible with the strata, (s₁, s₂) is aJ^′-holomorphic sec- tion ofY_χfor an almost complex structureJ^′such thatp_χ: Y_χ→X_r(f) is (J^′, J)-holomorphic. Now where ˜∆ ={(D₁, D₂)∈Y_χ|D₁∩D₂ 6=∅}, compatibility with the strata implies that ˜∆ will beJ^′-holomorphic. In a neighborhoodV around (s1(z), s2(z)), we have, in appropriate coordi- nates, ˜∆∩V ={(z, w, w, D₁, D₂)|w∈Σ_z},while (s₁(z), s₂(z)) has first three coordinates (z, g1(z), g2(z)). From this it follows by Lemma 3.4 of [IP2] that

g₁(z)−g₂(z) =a₀z^d+O(d+ 1)

for some d, in which case Cs1 and Cs2 have intersection multiplicity

d >0 atp. q.e.d.

Definition 2.6. Let Ω be a set of P

d(α_i) points and let J be an almost complex structure compatible with the strata. M^J,Ω₀ (α₁, . . . , α_n) shall denote the set of J-holomorphic sections s ∈ c_α with Ω ⊂ C_s such that there exist C¹ sections si ∈ cαi with s = P

si, while the s_i themselves do not admit nontrivial decompositions as sums of C¹ sections.

We would like to assert that for genericJ and Ω, the spaceM^J,Ω₀ (α1, . . . , α_n) does not include any sections contained within the strata. This is not true in full generality; rather we need the following assumption in order to rule out the effects of multiple covers of square-zero tori and (−1)-spheres in X.

Assumption 2.7. None of theαi can be written as αi =mβ where m >1 and either β² =κ_X·β= 0 orβ² =κ_X ·β=−1.

Under this assumption, we note that ifs=P

si ∈ M^J,Ω₀ (α1, . . . , αn) were contained in ∆, then since the α_i and hence the s_i are distinct we can write each s_i as s_i = m_is˜_i with at least one m_i > 1. The minimal stratum ofswill then beχ_π whereπ =n

r=P m_i³

ri

mi

´oand s^′ = ( ˜s₁, . . . ,s˜_n) will be a J^′-holomorphic section ofY_χ withs=p_χ◦s^′, in the homotopy class [c_α1/m1× · · · ×c_αn/mn].

If any of thed(α_i/m_i)<0, then Lemma 2.4 implies that there will be no such sections s^′ at all; otherwise (again by Lemma 2.4) the real di- mension of the space of such sections (taking into account the incidence conditions) will be

(2.1) 2³X

d(α_i/m_i)−X d(α_i)´

.

But an easy manipulation of the general formula for d(β) and the ad- junction formula (which applies here because the standard surface cor- responding to a section of X_r(f) which meets ∆ positively will be sym- plectic; c.f. Lemma 2.8 of [DS]) shows that if d(β)≥0 andm≥2 then d(mβ)> d(β) unless either β² =κ_X ·β = 0 or β² =κ_X ·β =−1, and these are ruled out in this context by (i) and (ii) above, respectively. So Assumption 2.7 implies that the dimension in Equation 2.1 is negative, so no such s^′ will exist for genericJ. This proves part of the following:

Proposition 2.8. Under Assumption 2.7, for generic pairs (J,Ω) where J is compatible with the strata and#Ω =P

d(α_i),M^J,Ω₀ (α₁, . . . , αn) is a finite set consisting only of sections not contained in ∆.

Proof. That no member ofM^J,Ω₀ (α₁, . . . , α_n) is contained in ∆ follows from the above discussion. As for the dimension of our moduli space, note that any s=P

s_i ∈ M^J,Ω₀ (α₁, . . . , α_n) has one tangency (counted with multiplicity) to ∆ for each of the intersections of the C_si, of which there are P

α_i ·α_j (counted with multiplicity; this multiplicity will

always be positive by Proposition 2.5). By the results of Section 6 of [IP2], the space M^J,Ω_δ,∆(c_α) of J-holomorphic sections in the class c_α having δ tangencies to ∆ and whose descendant surfaces pass through Ω will, for generic (J,Ω), be a manifold of dimension

2(d(α)−X

d(α_i)−δ) = 2³X

α_i·α_j −δ´ , which is equal to zero in the case δ =P

α_i·α_j of present relevance to us.

Let us now show thatM^J,Ω₀ (α₁, ..., α_n) is compact. Now since +(c_α1×

· · · ×c_αn) is C⁰-closed in c_α, by Gromov compactness any sequence s^(m) = Pn

i=1s^(m)_i in M^J,Ω₀ (α₁, . . . , α_n) has (after passing to a subse- quence) a J-holomorphic limit s=P

s_i where the s_i∈c_αi are at least continuous. We claim that, at least for generic (J,Ω), we can guarantee the s_i to be C¹. In light of Proposition 2.2, the differentiability of the s_i is obvious at all points wheresmisses the diagonal, sincesis smooth by elliptic regularity and the divisor addition map induces an isomor- phism on the tangent spaces away from the diagonal. Now each s^(m) hasP

α_i·α_j tangencies to the diagonal, corresponding to pointst∈S² at which some pair of the divisors s^(m)_i (t) share a point in common.

The limit s will then likewise have n tangencies to the diagonal; the dimension formulas in [IP2] ensure that for generic (J,Ω) no two of the tangencies will coalesce into a higher order tangency to the smooth part of ∆ in the limit, and all of the intersecions ofIm swith the smooth part of the diagonal other than these n tangencies will be transverse. Fur- thermore, one may easily show (using for instance an argument similar to the one used in Lemma 2.1 of [U1] to preclude generic 0-dimensional moduli spaces of pseudoholomorphic curves in a Lefschetz fibration from meeting the critical points) that since the singular locus of ∆ has codi- mension 4 inX_r(f), ifJhas been chosen generically thenswill not meet

∆^sing, and so no s(t) will contain more than one repeated point (and that point cannot appear with multiplicity larger than two). In light of this, each tangency of s to ∆ will occur at a point s(t) where some pair s_i(t) and s_j(t) have some point p in common, and all other points contained in anys_k(t) are distinct from each other and fromp. Thanks to Proposition 2.2, this effectively reduces us to the case r = 2, with s=s₁+s₂ a sum of continuous sections with s₁(0) =s₂(0) = 0 which is holomorphic with respect to an almost complex structure which pre- serves the diagonal stratum ∆ inD²×Sym²D², such thatsis tangent to ∆. Then lettingδ(t) = (s1+s2)²(t)−4s1(z)s2(t) be the discriminant, that s is tangent to the diagonal stratum implies, using Lemma 3.4 of [IP2], that δ(t) = at² +O(3) for some constant a; in particular δ(t) has twoC¹ square roots±r(t). Sincesis smooth, so is its first coordi- natet7→s₁(t) +s₂(t); adding this smooth function to the C¹ functions